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العربية

prove (3csc(-x))/(sec(-x))=-3cot(x)

الحلّ

prove

\frac{3 csc ( - x )}{sec ( - x )} = - 3 cot (x)

صحيح

خطوات الحلّ

\frac{3 csc ( - x )}{sec ( - x )} = - 3 cot (x)

قم بالعمل على الطرف الأيسر

\frac{3 csc ( - x )}{sec ( - x )}

csc (- x) = - csc (x)

:Use the negative angle identity

= \frac{3 ( - csc ( x ) )}{sec ( - x )}

sec (- x) = sec (x)

:Use the negative angle identity

= \frac{3 ( - csc ( x ) )}{sec ( x )}

sin, cos :

عبّر بواسطة

\frac{( - csc ( x ) ) \cdot 3}{sec ( x )}

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

= \frac{( - \frac{1}{s i n ( x )} ) \cdot 3}{sec ( x )}

sec (x) = \frac{1}{cos ( x )}

:Use the basic trigonometric identity

= \frac{( - \frac{1}{s i n ( x )} ) \cdot 3}{\frac{1}{c o s ( x )}}

\frac{( - \frac{1}{s i n ( x )} ) \cdot 3}{\frac{1}{c o s ( x )}}

بسّط:

- \frac{3 cos ( x )}{sin ( x )}

\frac{( - \frac{1}{s i n ( x )} ) \cdot 3}{\frac{1}{c o s ( x )}}

(- a) = - a

:احذف الأقواس

= \frac{- \frac{1}{s i n ( x )} \cdot 3}{\frac{1}{c o s ( x )}}

\frac{- a}{b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{\frac{1}{s i n ( x )} \cdot 3}{\frac{1}{c o s ( x )}}

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

: استخدم ميزات الكسور التالية

\frac{\frac{1}{s i n ( x )} \cdot 3}{\frac{1}{c o s ( x )}} = \frac{\frac{1}{s i n ( x )} \cdot 3 cos ( x )}{1}

= - \frac{\frac{1}{s i n ( x )} \cdot 3 cos ( x )}{1}

\frac{a}{1} = a

: استخدم ميزات الكسور التالية

\frac{\frac{1}{s i n ( x )} \cdot 3 cos ( x )}{1} = \frac{1}{sin ( x )} \cdot 3 cos (x)

= - \frac{1}{sin ( x )} \cdot 3 cos (x)

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= - \frac{1 \cdot 3 cos ( x )}{sin ( x )}

1 \cdot 3 = 3 :

اضرب الأعداد

= - \frac{3 cos ( x )}{sin ( x )}

= - \frac{3 cos ( x )}{sin ( x )}

= - \frac{3 cos ( x )}{sin ( x )}

Rewrite using trig identities

= - 3 \cdot \frac{cos ( x )}{sin ( x )}

\frac{cos ( x )}{sin ( x )} = cot (x)

:Use the basic trigonometric identity

- 3 cot (x)

- 3 cot (x)

أظهرنا بأنّ الطرفين من الممكن أن تكون لهما نفس الصورة

\Rightarrow صحيح

p ro v e \frac{1 - tan ^{4} ( x )}{1 - tan ^{2} ( x )} = sec^{2} (x)

p ro v e cos (4 x) = 1 - 2 sin^{2} (2 x)

p ro v e arcsin (x) = csc (x)

p ro v e cos^{4} (A) - sin^{4} (A) = cos (2 A)

p ro v e \frac{1 + sec ^{2} ( x )}{sec ^{2} ( x )} = 1 + cos^{2} (x)